A study of the second-order dynamical systems for variational inequalities

Abstract

The purpose of this paper is to study the variational inequality problem through a second order dynamical system. The dynamical system is a second-order dynamical system that involves the asymptotic vanishing of damping and Hessian-damping terms. Our approach primarily uses the asymptotic dynamics of the inertial system and the geometric characteristics of the damping factors. The asymptotic vanishing damping is directly linked to the method of the Nesterov accelerated gradient. The geometry of the Hessian damping term is in this form d dt(y(t)−w(t)), where y(t) is the generated trajectories by our proposed dynamical system, w(t) = PC(y(t)−Qy(t)) and Q is our operator which is strongly monotone. The Hessian term assists with the neutralisation of the oscillation. This combination forms a damping force that assists with controlling the convergence speed of dynamical systems, stabilizing the system, and mostly ensuring we have a controlled velocity over time. As a main result, we establish the existence and uniqueness of the generated trajectories using the Cauchy-Lipschitz theorem. We give two numerical non-Hilbert examples. We construct a Lyapunov function to achieve weak convergence, assuming that the underlying operator is Lipschitz continuous and monotone. By considering the strong monotonicity of the operator, we establish exponential convergence. Furthermore, we given two numerical examples to show the applicability of our results. Finally, we conduct numerical experiments to show the performance of our dynamical system. The results presented in this paper build upon and significantly enhance recent findings in the existing literature.